On hierarchical structures and reiterated homogenization

Ćwikel, Michael; Engliš, Miroslav; Kufner, A.; Persson, Lars‐Erik; Sparr, Gunnar

doi:10.1515/9783110198058.355

Cited by 4 publications

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

On the Elastic Deformation of Symmetric Periodic Structures

Lukkassen¹

2003

The Quarterly Journal of Mechanics and Applied Mathematics

View full text Add to dashboard Cite

SummaryWe discuss locally anisotropic periodic elastic materials in R n which are symmetric with respect to a hyperplane. The cell problem for finding the effective elastic parameters is considered. It is shown that the components of the displacement satisfy either Neumann or Dirichlet conditions on the sides of the cell of periodicity parallel with the symmetry plane. We also prove that the corresponding homogenized material is monocyclic with respect to this plane. The proof for the general case is based on variational and weak formulations of the elasticity problem. For the special case of plates with a square array of holes or perfectly-bonded inclusions we present an independent proof based on Kolosov-Muskhelishvili analytic potentials. In computational practice the derived boundary conditions permit us to easily incorporate the periodic boundaryvalue problem into standard numerical schemes.

show abstract

On the Elastic Deformation of Symmetric Periodic Structures

Lukkassen¹

2003

The Quarterly Journal of Mechanics and Applied Mathematics

View full text Add to dashboard Cite

show abstract

Multiscale homogenization of monotone operators

Lukkassen¹,

Meidell²,

Wall³

2008

Discrete &Amp; Continuous Dynamical Systems - A

View full text Add to dashboard Cite

show abstract

Iterated averaging of three-scale oscillatory systems

Ariel

Engquist

Kim

et al. 2014

Communications in Mathematical Sciences

View full text Add to dashboard Cite

A theory of iterated averaging is developed for a class of highly oscillatory ordinary differential equations (ODEs) with three well separated time scales. The solutions of these equations are assumed to be (almost) periodic in the fastest time scales. It is proved that the dynamics on the slowest time scale can be approximated by an effective ODE obtained by averaging out oscillations. In particular, the effective dynamics of the considered class of ODEs is always deterministic and does not show any stochastic effects. This is in contrast to systems in which the dynamics on the fastest time scale is mixing. The systems are studied from three perspectives: first, using the tools of averaging theory; second, by formal asymptotic expansions; and third, by averaging with respect to fast oscillations using nested convolutions with averaging kernels. The latter motivates a hierarchical numerical algorithm consisting of nested integrators.

show abstract

On hierarchical structures and reiterated homogenization

Cited by 4 publications

References 39 publications

On the Elastic Deformation of Symmetric Periodic Structures

On the Elastic Deformation of Symmetric Periodic Structures

Multiscale homogenization of monotone operators

Iterated averaging of three-scale oscillatory systems

Contact Info

Product

Resources

About